Conditioning 3D Stochastic Channels to Pressure Data
- Bi Zhuoxin (U. of Tulsa) | D.S. Oliver (U. of Tulsa) | A.C. Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2000
- Document Type
- Journal Paper
- 474 - 484
- 2000. Society of Petroleum Engineers
- 5.1 Reservoir Characterisation, 2.4.3 Sand/Solids Control, 5.5 Reservoir Simulation, 5.6.4 Drillstem/Well Testing, 1.2.3 Rock properties, 1.10 Drilling Equipment, 5.1.5 Geologic Modeling, 5.1.9 Four-Dimensional and Four-Component Seismic, 5.5.8 History Matching
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This work discusses the development and implementation of a procedure to condition a stochastic channel to well-test pressure data and well observations of the channel thickness and the depth of the top of the channel. The stochastic channel is defined by a set of geometric random variables (referred to as geometric model parameters) that describe the location, size, and shape. Channel and nonchannel permeability and porosity are treated as random variables. These four random variables plus the geometric parameters comprise the complete set of model parameters. Multiple conditional realizations of the geometric parameters and rock properties are generated to evaluate the uncertainty in model parameters and the reduction in uncertainty obtained by conditioning to well-test pressure data.
Georgsen and Omre1 have presented a stochastic model for river beds (referred to here as channels) within a channel belt and used this model to simulate a fluvial reservoir consisting of multiple channel belts. Except for the use of aspect ratio in place of thickness as one of the Gaussian random fields used to define the size of a stochastic channel, the set of geometric parameters that describe our channels is nearly identical to theirs. Georgsen and Omre also implemented a simulation algorithm to generate conditional realizations of multiple channels within a channel belt where the conditioning data consist of facies observations at each well. Georgsen2 extended the model of Georgsen and Omre to include four facies, namely, a background facies, sheet splay, channel sand, and barriers. In the simulation process the last three facies are embedded as geometric, geological objects within the background facies. All facies exist within a rectangular three-dimensional box which represents the complete reservoir system. In our work, we simulate a single channel within a rectangular three-dimensional box. However, unlike the two references cited above, the permeability and porosity within and outside the channel are considered to be random variables, so the model parameters include both the geometric parameters for the stochastic channel and the rock property fields. Moreover, we generate realizations of the model conditioned to single-phase flow pressure data as well as well observations of the channel.
The problem of conditioning a channel to pressure data has been considered previously by Landa and Horne3 (also Ref. 4) and Rahon et al.5 Landa and Horne, however, only considered two-dimensional channels and used a model in which the channel width is constant and channel boundaries are described by trigonometric functions, so that the number of channel geometric parameters was very small. Moreover, channel and nonchannel permeability and porosity were assumed to be known. They showed that the sensitivity of pressure to channel parameters can be calculated efficiently using the direct method. (The direct method is sometimes referred to as the sensitivity coefficient method6 in the hydrology literature and is commonly referred to as the gradient simulator method of Anterion et al.7 in the petroleum engineering literature.) In the work of Landa4 and Landa and Horne,3 channel parameters were estimated by applying the Gauss-Newton method to minimize an objective function consisting of a sum of squares of data mismatch terms. Various modifications, including Levenberg-Marquardt and modified Cholesky decomposition, were used as necessary to provide regularization. In the single synthetic example presented by Landa,4 the channel parameters were estimated using two-phase (oil-water) flow production data at multiple wells, drill stem test pressure data, and a map of changes in saturation. (Landa assumed the saturation map could be obtained reliably from four-dimensional seismic data.) For this example of Landa,4 the algorithm required 41 iterations to converge. He attributed the slow convergence to the existence of a local minimum and suggested reweighting of individual data mismatch terms in the objective function to overcome the problem. However, in the probabilistic approach applied here, the objective function to be minimized appears as the exponent of a probability density function and cannot be adjusted arbitrarily. For the problems we have considered, we found that convergence properties of the algorithm can sometimes be significantly improved by conditioning to an observation of the channel at the well. We present a procedure to incorporate such information in an efficient way that improves the convergence properties of the algorithm and is consistent with the randomized maximum likelihood method used to generate realizations.
Rahon et al.5 modeled a three-dimensional channel by triangularization of its bounding surfaces. In the three-dimensional example they considered, the channel is actually defined by a triangularized surface with 90 nodes. They consider only the case where the channel lies along a specified line, i.e., the centerline of the channel is a known line. They assume porosity and permeability within and outside the channel are known. Sensitivities are computed by a continuous form of the adjoint method and they estimate channel parameters (the positions of nodes of triangles) by minimizing an objective function that includes the squares of both pressure and pressure derivative data mismatches.
In conditioning a geostatistical model to dynamic data, computational efficiency is a key issue. Any method for generating conditional realizations of the model or the maximum a posteriori estimate requires the minimization of an appropriate objective function. Although many references have applied a Gauss-Newton method for optimization, one sometimes finds that it is necessary to damp the change in model parameters at early iterations to avoid slow convergence or convergence to a local minimum which gives an unacceptable match of observed data. This problem has been discussed recently by Wu et al.8 and is similar to convergence difficulties observed in very early work on automatic history matching; see, for example, Jacquard and Jain9 and Jahns.10 In this work, we use a Levenberg-Marquardt algorithm to provide this damping. Convergence is further accelerated by a special procedure for generating an initial guess of model parameters. For a Gauss-Newton or Levenberg-Marquardt algorithm to be efficient, it must converge rapidly and one must employ an efficient algorithm for computing the derivatives (sensitivity coefficients) needed to form the Hessian or modified Hessian matrix. As will be seen, our form of the Levenberg-Marquardt algorithm typically requires on the order of five iterations to obtain convergence and our method for generating sensitivity coefficients is computationally efficient.
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